• 대한수학회논문집
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• 제13권1호
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• pp.73-76
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• 1998

It is proved that the partial differential operator $D_x + ix^q D^2_y$ is not locally solvable in any open set which intersects the line x = 0, when $q = - \frac{2l-1}{2k-1}$ is not an integer.

• 통합자연과학논문집
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• 제7권4호
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• pp.273-277
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• 2014

In this article, we consider a singular and a degenerate elliptic operators in a divergence form. The singularities exist on a part of boundary, and comparable to the logarithmic distance function or its inverse. If we assume that the operator can be treated in a pointwise sense than distribution sense, with this operator we obtain a priori Harnack continuity near the boundary. In the proof we transform the singular elliptic operator to uniformly bounded elliptic operator with unbounded first order terms. We study this type of estimations considering a De Giorgi conjecture. In his conjecture, he proposed a certain ellipticity condition to guarantee a continuity of a solution.